Bayesian nonparametric factorizations of matrix-valued parameters

Matrix-valued parameters are often encountered in several statistical models. In modern applications, such matrices are increasingly high-dimensional and often sparse, thus motivating lower dimensional representations based on latent variables. A common problem within this context is the lack of knowledge on the dimension of the latent space, which has to be learned from the observed data. To address this issue, it is possible to rely either on over-complete representations in combination with suitable shrinkage priors, or to consider a prior distribution also for the unknown dimension of the latent space. In this talk, we discuss how Bayesian nonparametric methods can be effective in both directions. More specifically, we first present a novel increasing shrinkage prior, named cumulative shrinkage process, which is based on a sequence of spike-and-slab distributions that assigns increasing mass to the spike as model complexity grows via a cumulative sum of stick-breaking probabilities. Using factor analysis as an illustrative example, we show that this formulation has theoretical and practical advantages, including an improved ability to recover the model dimension. In the second part of the talk, we focus instead on the problem of learning block-structures among groups of nodes in network data, while recovering the unknown number of clusters. Motivated by criminal network studies, we propose a general framework for stochastic block modeling, named extended stochastic block model, that infers groups of nodes via Gibbs–type priors on the partition process. Within this unified formulation, we focus in particular on the Gnedin process as a realistic prior that allows the number of groups to be finite, random and subject to a reinforcement process coherent with the modular structures in organized crime. The performance is illustrated in an application to an Italian Mafia network, where we unveil complex block structures, mostly hidden from state-of-the-art alternative solutions.

Joint works with David B. Dunson and Tommaso Rigon